Loading...
The normal distribution (bell curve) is the most important probability distribution in statistics. It describes data that clusters around a mean, with values becoming less frequent the farther they are from the center. About 68% of data falls within 1 standard deviation, 95% within 2, and 99.7% within 3 (the Empirical Rule).
Z-Score Formula (z)
z = (x - μ) / σ
These define your normal distribution. Example: Test scores with μ = 75 and σ = 10.
μ = mean, σ = standard deviationSubtract the mean from your value, then divide by the standard deviation. For x = 85: z = (85 − 75) / 10 = 1.0. This means 85 is exactly 1 standard deviation above the mean.
z = (x - μ) / σA z-score of 1.0 corresponds to a cumulative probability of 0.8413, meaning 84.13% of values fall below 85. For the probability above 85: 1 − 0.8413 = 0.1587 (15.87%).
P(X < x) = Φ(z)To find P(70 < X < 85): z₁ = (70−75)/10 = −0.5, z₂ = (85−75)/10 = 1.0. P = Φ(1.0) − Φ(−0.5) = 0.8413 − 0.3085 = 0.5328 (53.28%).
P(a < X < b) = Φ(z₂) − Φ(z₁)68-95-99.7 Rule: In any normal distribution, 68.27% of values fall within ±1σ of the mean, 95.45% within ±2σ, and 99.73% within ±3σ. This is a quick way to estimate probabilities without a z-table.
Normal distributions appear in heights, test scores, measurement errors, blood pressure, IQ scores, manufacturing tolerances, and many natural phenomena. If a process is influenced by many small, independent factors, the result tends to be normally distributed (Central Limit Theorem).
Remember that the z-table gives cumulative probability from the LEFT (P(X < x)). For P(X > x), subtract from 1. For a range, subtract the smaller cumulative probability from the larger one.
Skip the manual calculations. Use our free Normal Distribution Calculator for instant results.
Open Normal Distribution Calculator →